Abstract

Let be a prime ring of characteristic different from 2, with extended centroid , its two-sided Utumi quotient ring, a nonzero generalized derivation of , a noncentral multilinear polynomial over in noncommuting variables, and such that for any . Then one of the following holds: (1) ; (2) ; (3) there exists such that , for all ; (4) there exist and such that , for all , and is central valued on ; (5) there exist and such that , for all , and , .

1. Introduction

Let be a prime ring with center . We denote by the simple commutator of the elements and by , for , the th commutator of . Throught this paper we will use the following notation: will be the (two-sided) Utumi quotient ring of a ring (sometimes, as in [1], is called the symmetric ring of quotients). The definition, the axiomatic formulation, and the properties of this quotient ring can be found in [1–3].

In any case, when is a prime ring, all that we need here about this object is that(1);(2) is a prime ring;(3)the center of , denoted by is a field which is called the extended centroid of .A well known result of Posner [4] says that if is a derivation of such that , for all , then is commutative. In [5] Lanski generalizes the result of Posner, by replacing the element with an element of a noncentral Lie ideal of . More precisely he proves that if for all and a fixed integer; then and satisfies , the standard identity of degree .

Let be a multilinear polynomial over in noncommuting variables and denote by the set of all evaluations of in . In case is not central valued on , it is well known that the additive subgroup generated by contains a noncentral Lie ideal of . Moreover any noncentral Lie ideal of contains all the commutators for in some nonzero ideal of , unless and .

In light of this and following the line of investigation of the previous cited papers, in [6] P. H. Lee and T. K. Lee consider the Engel-condition , in case , where is a two-sided ideal of . They show that either is central valued in or and satisfies .

These results indicate that the global structure of a prime ring is often tightly connected to the behaviour of additive mappings defined on , which act on suitable subsets of the whole ring. In [7] de Filippis and di Vincenzo study the left annihilator of the set , where is a derivation. In case the annihilator is not zero, the conclusion is that is central valued on . These facts in a prime ring are natural tests which evidence that the set is rather large in .

More recently, Liu [8] and Wang [9] have examined the identity , where is a derivation of and , where is a one-sided ideal of . In particular, for , if and is not central valued on , then and satisfies .

In [10] de Filippis considers a similar situation, in the case the derivation is replaced by a generalized derivation . An additive map is said to be a generalized derivation if there is a derivation of such that, for all , . A significative example is a map of the form , for some ; such generalized derivations are called inner. Generalized derivations have been primarily studied on operator algebras. Therefore any investigation from the algebraic point of view might be interesting (see, e.g., [11]).

The main result in [10] is the following.

Theorem A. Let be a prime ring of characteristic different from 2, with extended centroid , its two-sided Utumi quotient ring, a nonzero generalized derivation of , a noncentral multilinear polynomial over in n noncommuting variables, and such that for any . Then either or one of the following holds: (1)there exists such that , for all ;(2)there exist and such that , for all , and is central valued on .

We would like to remark that the same conclusions hold in case we consider the right annihilator, more precisely.

Theorem B. Let be a prime ring of characteristic different from 2, with extended centroid , its two-sided Utumi quotient ring, a nonzero generalized derivation of , a noncentral multilinear polynomial over in n noncommuting variables, and such that for any . Then either or one of the following holds: (1)there exists such that , for all ;(2)there exist and such that , for all , and is central valued on .

Here we will consider a more general situation, involving a two-sided annihilating condition. More specifically, we study simultaneously left and right annihilators of the set and prove the following.

Theorem 1. Let be a prime ring of characteristic different from 2, with extended centroid , its two-sided Utumi quotient ring, a nonzero generalized derivation of , a noncentral multilinear polynomial over in n noncommuting variables, and such that for any . Then one of the following holds: (1);(2);(3)there exists such that , for all ;(4)there exist and such that , for all , and is central valued on ;(5)there exist and such that , for all , and , .

Remark 2. By the primeness of and in light of Theorems A and B, we may assume that is not a domain. Moreover, since the center of a prime ring cannot contain nonzero zero-divisor, then neither nor . Finally in all that follows we always suppose .

In the sequel we will make a frequent use of the following.

Remark 3. If is a basis of over then any element of , the free product over of the -algebra and the free -algebra , is called generalized polynomial and can be written in the form . In this decomposition the coefficients are in and the elements are -monomials; that is, , with and . In [12] it is shown that a generalized polynomial is the zero element of if and only if all are zero. Let be linearly independent over and , for some . If, for any , and , then are the zero element of . The same conclusion holds if , and for some .
We refer the reader to [1, 12] for more details on generalized polynomial identities.

2. An Independent Result

We will dedicate this section to the proof of the following proposition on linear identities with commutators in matrix rings. This result will be useful in the sequel.

Proposition 4. Let be a field and the algebra of matrices over and . Let , such that and for all . Then there exists such that and .

In order to prove Proposition 4, we need several lemmas.

Lemma 5. Let be an infinite field and . If are not scalar matrices in , then there exists some invertible matrix such that each matrix has all nonzero entries.

Proof. See Lemma  1.5 in [13].

Lemma 6. Let be a prime ring with extended centroid . Suppose , for all , where , for and . If are -independent then each is -dependent on . Analogously if are -independent then each is -dependent on .

Proof. It is Martindale’s result contained in [14].

Lemma 7. Let be a prime ring with extended centroid . Suppose , for all , where . Then .

Proof. It is an easy consequence of Lemma 6.

Lemma 8. Let be an infinite field, the algebra of matrices over , the center of , and . Assume that there exist nonzero elements of such that for all . If then one of the following holds: (1) are central matrices and ;(2) is a central matrix and .

Proof. Since , by the assumption, we have that for all . Clearly if then for all , and by Lemma 7 we get ; that is, . On the other hand, if , then for all and it follows easily that .
In light of this, we consider and both nonscalar matrices. We will prove that in this case we get a contradiction.
Here we denote by the usual matrix unit with in the -entry and zero elsewhere.
By Lemma 5, we can assume that and have all nonzero entries, say and , for , .
Since for all , then, for any , in particular the -entry of is , a contradiction.

For sake of clearness, we may write the previous lemma as follows.

Lemma 9. Let be an infinite field, the algebra of matrices over , the center of , and . Let be nonzero elements of such that for all . Assume there exists such that . Then and , for a suitable .

Lemma 10. Let be an infinite field, the algebra of matrices over , and the center of . Assume that there exist nonzero elements of such that for all . If and , for a suitable , then .

Proof. Assume that is not a scalar matrix. By Lemma 5, we can assume that and have all nonzero entries, say and , for , .
Since , for a suitable , by our assumption we have that that is, for all . In particular for , with , By calculations one has that the -entry of is , a contradiction.
Therefore must be a central matrix. In light of this, there exist such that and , so that , for all . Once again by Lemma 7 and since , it follows that ; that is, and .

Lemma 11. Let be an infinite field, the algebra of matrices over , and . Suppose there exist such that for all . Denote for suitable , and elements of . If there are such that , , and , then and for all and (i.e., the only nonzero off-diagonal elements of fall in the th row).

Proof. Consider the assumption In particular, for , we have so that, for all , the -entry of the matrix is . Since , one has for all , in particular . Thus, in case we are done (since ).
Assume in what follows that , and choose , with . Hence we also have From the previous equalities it follows that(1)for all , the -entry of the matrix is ;(2)for all , the -entry of the matrix is ;(3)the -entry of the matrix is ;(4)for all , the -entry of the matrix is (note that this holds also in case ).By and and since and , we have both , for all , and for all and . So by for all . Finally by , for all and .

Lemma 12. Let be an infinite field, the algebra of matrices over , and . Suppose there exist such that for all . Denote for suitable , , and elements of . Assume there are such that . If , for all , then one of the following holds: (1);(2), , and there exists such that

Proof. Firstly we consider the case . The first step is to apply twice Lemma 11: this forces to be a diagonal matrix. In fact , , and imply that for all and ; in particular, since , there exists such that , for all . Since , , we have for all and , so for all , as required. Say .
Consider now the inner automorphism of induced by the invertible matrix , for : . Of course , for all . Moreover the -entries of , , and are, respectively, , , and . Therefore, again by Lemma 11, any -entry of is zero, for all . By calculations ; that is, .
On the other hand, if is the inner automorphisms induced by the invertible matrix , as above , for all . Since the -entries of , , and are, respectively, , , and , and again any -entry of is zero, for all ; that is, and , for all . Thus is a central matrix in . By Lemma 9, either for some or . Since the first case cannot occur, we get and also which follows from and .
Let now ; that is, . In this case it is well known that for any element there exist such that . Without loss of generality we may assume . In case , then by the same above argument we show that and we are done again. Thus we consider the case . Moreover, by applying Lemma 11 it follows . Hence we may write For we have so that the -entry of the matrix is ; that is, and the -entry of the matrix is ; that is, . On the other hand, for , we have The -entry of the matrix is ; that is, and . Moreover the -entry of the matrix is . Therefore, if denoted , one has and .
Analogously, the -entry of the matrix is . Thus and .
Finally, by our assumption and for , with , we also have and by easy calculations it follows .